{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Table of Contents](./table_of_contents.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Designing Nonlinear Kalman Filters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import division, print_function\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <style>\n",
       "        .output_wrapper, .output {\n",
       "            height:auto !important;\n",
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       "        .output_scroll {\n",
       "            box-shadow:none !important;\n",
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       "        </style>\n",
       "    "
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#format the book\n",
    "import book_format\n",
    "book_format.set_style()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Author's note: I was initially planning to have a design nonlinear chapter that compares various approaches. This may or may not happen, but for now this chapter has no useful content and I suggest not reading it. **"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the Kalman filter reasonably tracks the ball. However, as already explained, this is a silly example; we can predict trajectories in a vacuum with arbitrary precision; using a Kalman filter in this example is a needless complication."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kalman Filter with Air Drag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I will dispense with the step 1, step 2, type approach and proceed in a more natural style that you would use in a non-toy engineering problem. We have already developed a Kalman filter that does excellently at tracking a ball in a vacuum, but that does not incorporate the effects of air drag into the model. We know that the process model is implemented with $\\textbf{F}$, so we will turn our attention to that immediately.\n",
    "\n",
    "Notionally, the computation that $\\textbf{F}$ computes is\n",
    "\n",
    "$$x' = Fx$$\n",
    "\n",
    "With no air drag, we had\n",
    "\n",
    "$$\n",
    "\\mathbf{F} = \\begin{bmatrix}\n",
    "1 & \\Delta t & 0 & 0 & 0 \\\\\n",
    "0 & 1 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 1 & \\Delta t & \\frac{1}{2}{\\Delta t}^2 \\\\\n",
    "0 & 0 & 0 & 1 & \\Delta t \\\\\n",
    "0 & 0 & 0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which corresponds to the equations\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "x &= x + v_x \\Delta t \\\\\n",
    "v_x &= v_x \\\\\n",
    "\\\\\n",
    "y &= y + v_y \\Delta t + \\frac{a_y}{2} {\\Delta t}^2 \\\\\n",
    "v_y &= v_y + a_y \\Delta t \\\\\n",
    "a_y &= a_y\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the section above we know that our new Euler equations must be\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "x &= x + v_x \\Delta t \\\\\n",
    "v_x &= v_x \\\\\n",
    "\\\\\n",
    "y &= y + v_y \\Delta t + \\frac{a_y}{2} {\\Delta t}^2 \\\\\n",
    "v_y &= v_y + a_y \\Delta t \\\\\n",
    "a_y &= a_y\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Realistic 2D Position Sensors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The position sensor in the last example are not very realistic. In general there is no 'raw' sensor that provides (x,y) coordinates. We have GPS, but GPS already uses a Kalman filter to create a filtered output; we should not be able to improve the signal by passing it through another Kalman filter unless we incorporate additional sensors to provide additional information. We will tackle that problem later. \n",
    "\n",
    "Consider the following set up. In an open field we put two transmitters at a known location, each transmitting a signal that we can detect. We process the signal and determine  how far we are from that signal, with some noise. First, let's look at a visual depiction of that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "circle1=plt.Circle((-4, 0), 5, color='#004080', \n",
    "                   fill=False, linewidth=20, alpha=.7)\n",
    "circle2=plt.Circle((4, 0), 5, color='#E24A33', \n",
    "                   fill=False, linewidth=5, alpha=.7)\n",
    "\n",
    "fig = plt.gcf()\n",
    "ax = fig.gca()\n",
    "\n",
    "plt.axis('equal')\n",
    "plt.xlim((-10, 10))\n",
    "plt.ylim((-10, 10))\n",
    "\n",
    "plt.plot ([-4, 0], [0, 3], c='#004080')\n",
    "plt.plot ([4, 0], [0, 3], c='#E24A33')\n",
    "plt.text(-4, -.5, \"A\", fontsize=16, horizontalalignment='center')\n",
    "plt.text(4, -.5, \"B\", fontsize=16, horizontalalignment='center')\n",
    "\n",
    "ax.add_artist(circle1)\n",
    "ax.add_artist(circle2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here I have attempted to show transmitter A, drawn in red, at (-4,0) and a second one B, drawn in blue, at (4,0). The red and blue circles show the range from the transmitters to the robot, with the width illustrating the effect of the $1\\sigma$ angular error for each transmitter. Here I have given the blue transmitter more error than the red one. The most probable position for the robot is where the two circles intersect, which I have depicted with the red and blue lines. You will object that we have two intersections, not one, but we will see how we deal with that when we design the measurement function.\n",
    "\n",
    "This is a very common sensor set up. Aircraft still use this system to navigate, where it is called DME (Distance Measuring Equipment). Today GPS is a much more common navigation system, but I have worked on an aircraft where we integrated sensors like this into our filter along with the GPS, INS, altimeters, etc. We will tackle what is called *multi-sensor fusion* later; for now we will just address this simple configuration.\n",
    "\n",
    "The first step is to design our state variables. We will assume that the robot is traveling in a straight direction with constant velocity. This is unlikely to be true for a long period of time, but is acceptable for short periods of time. This does not differ from the previous problem - we will want to track the values for the robot's position and velocity. Hence,\n",
    "\n",
    "$$\\mathbf{x} = \n",
    "\\begin{bmatrix}x\\\\v_x\\\\y\\\\v_y\\end{bmatrix}$$\n",
    "\n",
    "The next step is to design the state transition function. This also will be the same as the previous problem, so without further ado,\n",
    "\n",
    "$$\n",
    "\\mathbf{x}' = \\begin{bmatrix}1& \\Delta t& 0& 0\\\\0& 1& 0& 0\\\\0& 0& 1& \\Delta t\\\\ 0& 0& 0& 1\\end{bmatrix}\\mathbf{x}$$\n",
    "\n",
    "The next step is to design the control inputs. We have none, so we set ${\\mathbf{B}}=0$.\n",
    "\n",
    "The next step is to design the measurement function $\\mathbf{z} = \\mathbf{Hx}$. We can model the measurement using the Pythagorean theorem.\n",
    "\n",
    "$$\n",
    "z_a = \\sqrt{(x-x_A)^2 + (y-y_A)^2} + v_a\\\\[1em]\n",
    "z_b = \\sqrt{(x-x_B])^2 + (y-y_B)^2} + v_b\n",
    "$$\n",
    "\n",
    "where $v_a$ and $v_b$ are white noise.\n",
    "\n",
    "We see an immediate problem. The Kalman filter is designed for linear equations, and this is obviously nonlinear. In the next chapters we will look at several ways to handle nonlinear problems in a robust way, but for now we will do something simpler. If we know the approximate position of the robot than we can linearize these equations around that point. I could develop the generalized mathematics for this technique now, but instead let me just present the worked example to give context to that development."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of computing $\\mathbf{H}$ we will compute the partial derivative of $\\mathbf{H}$ with respect to the robot's position $\\mathbf{x}$. You are probably familiar with the concept of partial derivative, but if not, it just means how $\\mathbf{H}$ changes with respect to the robot's position. It is computed as the partial derivative of $\\mathbf{H}$ as follows:\n",
    "\n",
    "$$\\frac{\\partial \\mathbf{h}}{\\partial \\mathbf{x}} = \n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial h_1}{\\partial x_1} & \\frac{\\partial h_1}{\\partial x_2} &\\dots \\\\\n",
    "\\frac{\\partial h_2}{\\partial x_1} & \\frac{\\partial h_2}{\\partial x_2} &\\dots \\\\\n",
    "\\vdots & \\vdots\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Let's work the first partial derivative. We want to find\n",
    "\n",
    "$$\\frac{\\partial }{\\partial x} \\sqrt{(x-x_A)^2 + (y-y_A)^2}\n",
    "$$\n",
    "\n",
    "Which we compute as\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{\\partial h_1}{\\partial x} &= ((x-x_A)^2 + (y-y_A)^2))^\\frac{1}{2} \\\\\n",
    "&= \\frac{1}{2}\\times 2(x-x_a)\\times ((x-x_A)^2 + (y-y_A)^2))^{-\\frac{1}{2}} \\\\\n",
    "&= \\frac{x_r - x_A}{\\sqrt{(x_r-x_A)^2 + (y_r-y_A)^2}} \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "We continue this computation for the partial derivatives of the two distance equations with respect to $x$, $y$, $dx$ and $dy$, yielding\n",
    "\n",
    "$$\\frac{\\partial\\mathbf{h}}{\\partial\\mathbf{x}}=\n",
    "\\begin{bmatrix}\n",
    "\\frac{x_r - x_A}{\\sqrt{(x_r-x_A)^2 + (y_r-y_A)^2}} & 0 & \n",
    "\\frac{y_r - y_A}{\\sqrt{(x_r-x_A)^2 + (y_r-y_A)^2}} & 0 \\\\\n",
    "\\frac{x_r - x_B}{\\sqrt{(x_r-x_B)^2 + (y_r-y_B)^2}} & 0 &\n",
    "\\frac{y_r - y_B}{\\sqrt{(x_r-x_B)^2 + (y_r-y_B)^2}} & 0 \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "That is pretty painful, and these are very simple equations. Computing the Jacobian can be extremely difficult or even impossible for more complicated systems. However, there is an easy way to get Python to do the work for you by using the SymPy module [1]. SymPy is a Python library for symbolic mathematics. The full scope of its abilities are beyond this book, but it can perform algebra, integrate and differentiate equations, find solutions to differential equations, and much more. We will use it to compute our Jacobian!\n",
    "\n",
    "First, a simple example. We will import SymPy, initialize its pretty print functionality (which will print equations using LaTeX). We will then declare a symbol for NumPy to use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAATBAMAAAC0B+rjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIokQdkQymVRmzbur3e+SS/cOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAYElEQVQIHWNgYGBgBGIgYEKhTCC8MAilAKLYAgsbgVQF4wQuAQbuCZwCTAcY5gK1sV9g2A6kuA4wvGSQZJBvYFjCMBEowhDLoMBdANRWuaUMpJ1xAohk4BQAU1DbOEEcAFCSDm4gARWbAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "\\phi"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy\n",
    "from sympy import init_printing\n",
    "init_printing(use_latex='png')\n",
    "\n",
    "phi, x = sympy.symbols('\\phi, x')\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how we use a latex expression for the symbol `phi`. This is not necessary, but if you do it will render as LaTeX when output. Now let's do some math. What is the derivative of $\\sqrt{\\phi}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       " 1  \n",
       "────\n",
       "2⋅√φ"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff('sqrt(phi)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can factor equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "        ⎛ 2    ⎞\n",
       "(φ - 1)⋅⎝φ  + 1⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor('phi**3 -phi**2 + phi - 1')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy has a remarkable list of features, and as much as I enjoy exercising its features we cannot cover them all here. Instead, let's compute our Jacobian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "⎡           x - xₐ                           y - yₐ              ⎤\n",
       "⎢ ──────────────────────────   0   ──────────────────────────   0⎥\n",
       "⎢    _______________________          _______________________    ⎥\n",
       "⎢   ╱         2           2          ╱         2           2     ⎥\n",
       "⎢ ╲╱  (x - xₐ)  + (y - yₐ)         ╲╱  (x - xₐ)  + (y - yₐ)      ⎥\n",
       "⎢                                                                ⎥\n",
       "⎢          x - x_b                          y - y_b              ⎥\n",
       "⎢────────────────────────────  0  ────────────────────────────  0⎥\n",
       "⎢   _________________________        _________________________   ⎥\n",
       "⎢  ╱          2            2        ╱          2            2    ⎥\n",
       "⎣╲╱  (x - x_b)  + (y - y_b)       ╲╱  (x - x_b)  + (y - y_b)     ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import symbols, Matrix\n",
    "phi = symbols('\\phi')\n",
    "phi\n",
    "\n",
    "x, y, xa, xb, ya, yb, dx, dy = symbols('x y x_a x_b y_a y_b dx dy')\n",
    "\n",
    "H = Matrix([[sympy.sqrt((x-xa)**2 + (y-ya)**2)], \n",
    "            [sympy.sqrt((x-xb)**2 + (y-yb)**2)]])\n",
    "\n",
    "state = Matrix([x, dx, y, dy])\n",
    "H.jacobian(state)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a nutshell, the entry (0,0) contains the difference between the x coordinate of the robot and transmitter A's x coordinate divided by the distance between the robot and A. (2,0) contains the same, except for the y coordinates of the robot and transmitters. The bottom row contains the same computations, except for transmitter B. The 0 entries account for the velocity components of the state variables; naturally the range does not provide us with velocity.\n",
    "\n",
    "The values in this matrix change as the robot's position changes, so this is no longer a constant; we will have to recompute it for every time step of the filter.\n",
    "\n",
    "If you look at this you may realize that this is just a computation of x/dist and y/dist, so we can switch this to a trigonometic form with no loss of generality:\n",
    "\n",
    "$$\\frac{\\partial\\mathbf{h}}{\\partial\\mathbf{x}}=\n",
    "\\begin{bmatrix}\n",
    "-\\cos{\\theta_A} & 0 & -\\sin{\\theta_A} & 0 \\\\\n",
    "-\\cos{\\theta_B} & 0 & -\\sin{\\theta_B} & 0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "However, this raises a huge problem. We are no longer computing $\\mathbf{H}$, but $\\Delta\\mathbf{H}$, the change of $\\mathbf{H}$. If we passed this into our Kalman filter without altering the rest of the design the output would be nonsense. Recall, for example, that we multiply $\\mathbf{Hx}$ to generate the measurements that would result from the given estimate of $\\mathbf{x}$ But now that $\\mathbf{H}$ is linearized around our position it contains the *change* in the measurement function. \n",
    "\n",
    "We are forced, therefore, to use the *change* in $\\mathbf{x}$ for our state variables. So we have to go back and redesign our state variables. \n",
    "\n",
    ">Please note this is a completely normal occurrence in designing Kalman filters. The textbooks present examples like this as *fait accompli*, as if it is trivially obvious that the state variables needed to be velocities, not positions. Perhaps once you do enough of these problems it would be trivially obvious, but at that point why are you reading a textbook? I find myself reading through a presentation multiple times, trying to figure out why they made a choice, finally to realize that it is because of the consequences of something on the next page. My presentation is longer, but it reflects what actually happens when you design a filter. You make what seem reasonable design choices, and as you move forward you discover properties that require you to recast your earlier steps. As a result, I am going to somewhat abandon my **step 1**, **step 2**, etc.,  approach, since so many real problems are not quite that straightforward."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If our state variables contain the velocities of the robot and not the position then how do we track where the robot is? We can't. Kalman filters that are linearized in this fashion use what is called a *nominal trajectory* - i.e. you assume a position and track direction, and then apply the changes in velocity and acceleration to compute the changes in that trajectory. How could it be otherwise? Recall the graphic showing the intersection of the two range circles - there are two areas of intersection. Think of what this would look like if the two transmitters were very close to each other - the intersections would be two very long crescent shapes. This Kalman filter, as designed, has no way of knowing your true position from only distance measurements to the transmitters. Perhaps your mind is already leaping to ways of working around this problem. If so, stay engaged, as later sections and chapters will provide you with these techniques. Presenting the full solution all at once leads to more confusion than insight, in my opinion. \n",
    "\n",
    "So let's redesign our *state transition function*. We are assuming constant velocity and no acceleration, giving state equations of\n",
    "$$\n",
    "\\dot{x}' = \\dot{x} \\\\\n",
    "\\ddot{x}' = 0 \\\\\n",
    "\\dot{y}' = \\dot{y} \\\\\n",
    "\\dot{y}' = 0$$\n",
    "\n",
    "This gives us the the *state transition function* of\n",
    "\n",
    "$$\n",
    "\\mathbf{F} = \\begin{bmatrix}0 &1 & 0& 0\\\\0& 0& 0& 0\\\\0& 0& 0& 1\\\\ 0& 0& 0& 0\\end{bmatrix}$$\n",
    "\n",
    "A final complication comes from the measurements that we pass in. $\\mathbf{Hx}$ is now computing the *change* in the measurement from our nominal position, so the measurement that we pass in needs to be not the range to A and B, but the *change* in range from our measured range to our nominal position. \n",
    "\n",
    "There is a lot here to take in, so let's work through the code bit by bit. First we will define a function to compute $\\frac{\\partial\\mathbf{h}}{\\partial\\mathbf{x}}$ for each time step."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sin, cos, atan2\n",
    "\n",
    "def H_of(pos, pos_A, pos_B):\n",
    "    \"\"\" Given the position of our object at 'pos' in 2D, and two \n",
    "    transmitters A and B at positions 'pos_A' and 'pos_B', return \n",
    "    the partial derivative of H\n",
    "    \"\"\"\n",
    "\n",
    "    theta_a = atan2(pos_a[1] - pos[1], pos_a[0] - pos[0])\n",
    "    theta_b = atan2(pos_b[1] - pos[1], pos_b[0] - pos[0])\n",
    "\n",
    "    return np.array([[0, -cos(theta_a), 0, -sin(theta_a)],\n",
    "                     [0, -cos(theta_b), 0, -sin(theta_b)]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we need to create our simulated sensor. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.random import randn\n",
    "\n",
    "class DMESensor(object):\n",
    "    def __init__(self, pos_a, pos_b, noise_factor=1.0):\n",
    "        self.A = pos_a\n",
    "        self.B = pos_b\n",
    "        self.noise_factor = noise_factor\n",
    "        \n",
    "    def range_of(self, pos):\n",
    "        \"\"\" returns tuple containing noisy range data to A and B\n",
    "        given a position 'pos'\n",
    "        \"\"\"\n",
    "        \n",
    "        ra = math.sqrt((self.A[0] - pos[0])**2 + (self.A[1] - pos[1])**2)\n",
    "        rb = math.sqrt((self.B[0] - pos[0])**2 + (self.B[1] - pos[1])**2)\n",
    "        \n",
    "        return (ra + randn()*self.noise_factor, \n",
    "                rb + randn()*self.noise_factor)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we are ready for the Kalman filter code. I will position the transmitters at x=-100 and 100, both with y=-20. This gives me enough space to get good triangulation from both as the robot moves. I will start the robot at (0,0) and move by (1,1) each time step. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import kf_book.book_plots as bp\n",
    "from filterpy.kalman import KalmanFilter\n",
    "import math\n",
    "import numpy as np\n",
    "\n",
    "pos_a = (100, -20)\n",
    "pos_b = (-100, -20)\n",
    "\n",
    "f1 = KalmanFilter(dim_x=4, dim_z=2)\n",
    "\n",
    "f1.F = np.array ([[0, 1, 0, 0],\n",
    "                  [0, 0, 0, 0],\n",
    "                  [0, 0, 0, 1],\n",
    "                  [0, 0, 0, 0]], dtype=float)\n",
    "\n",
    "f1.R *= 1.\n",
    "f1.Q *= .1\n",
    "\n",
    "f1.x = np.array([[1, 0, 1, 0]], dtype=float).T\n",
    "f1.P = np.eye(4) * 5.\n",
    "\n",
    "# initialize storage and other variables for the run\n",
    "count = 30\n",
    "xs, ys = [], []\n",
    "pxs, pys = [], []\n",
    "\n",
    "# create the simulated sensor\n",
    "d = DMESensor(pos_a, pos_b, noise_factor=3.)\n",
    "\n",
    "# pos will contain our nominal position since the filter does not\n",
    "# maintain position.\n",
    "pos = [0, 0]\n",
    "\n",
    "for i in range(count):\n",
    "    # move (1,1) each step, so just use i\n",
    "    pos = [i, i]\n",
    "    \n",
    "    # compute the difference in range between the nominal track\n",
    "    # and measured ranges\n",
    "    ra,rb = d.range_of(pos)\n",
    "    rx,ry = d.range_of((pos[0] + f1.x[0, 0], pos[1] + f1.x[2, 0]))\n",
    "    z = np.array([[ra - rx], [rb - ry]])\n",
    "\n",
    "    # compute linearized H for this time step\n",
    "    f1.H = H_of (pos, pos_a, pos_b)\n",
    "\n",
    "    # store stuff so we can plot it later\n",
    "    xs.append(f1.x[0, 0]+i)\n",
    "    ys.append(f1.x[2, 0]+i)\n",
    "    pxs.append(pos[0])\n",
    "    pys.append(pos[1])\n",
    "    \n",
    "    # perform the Kalman filter steps\n",
    "    f1.predict()\n",
    "    f1.update(z)\n",
    "\n",
    "bp.plot_filter(xs, ys)\n",
    "bp.plot_track(pxs, pys)\n",
    "plt.legend(loc=2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linearizing the Kalman Filter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have seen an example of linearizing the Kalman filter we are in a position to better understand the math. \n",
    "\n",
    "We start by assuming some function $\\mathbf f$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: A falling Ball"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**author's note: ignore this section for now. **\n",
    "\n",
    "In the **Designing Kalman Filters** chapter I first considered tracking a ball in a vacuum, and then in the atmosphere. The Kalman filter performed very well for vacuum, but diverged from the ball's path in the atmosphere. Let us look at the output; to avoid littering this chapter with code from that chapter I have placed it all in the file `ekf_internal.py'."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 900x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import kf_book.ekf_internal as ekf\n",
    "ekf.plot_ball()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can artificially force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
    "\n",
    "Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
    "\n",
    "$$a_x = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_x \\\\\n",
    "a_y = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_y- g\n",
    "$$\n",
    "\n",
    "These equations will be *very* unpleasant to work with while we develop this subject, so for now I will retreat to a simpler one dimensional problem using this simplified equation for acceleration that does not take the nonlinearity of the drag coefficient into account:\n",
    "\n",
    "\n",
    "$$\\begin{aligned}\n",
    "\\ddot{y} &= \\frac{0.0034ge^{-y/20000}\\dot{y}^2}{2\\beta} - g \\\\\n",
    "\\ddot{x} &= \\frac{0.0034ge^{-x/20000}\\dot{x}^2}{2\\beta}\n",
    "\\end{aligned}$$\n",
    "\n",
    "Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is still nonlinear, so we need to linearize this equation at the current state point. If our state is position and velocity, we need an equation for some arbitrarily small change in $\\mathbf{x}$, like so:\n",
    "\n",
    "$$ \\begin{bmatrix}\\Delta \\dot{x} \\\\ \\Delta \\ddot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix} = \n",
    "\\large\\begin{bmatrix}\n",
    "\\frac{\\partial \\dot{x}}{\\partial x} & \n",
    "\\frac{\\partial \\dot{x}}{\\partial \\dot{x}} & \n",
    "\\frac{\\partial \\dot{x}}{\\partial y} & \n",
    "\\frac{\\partial \\dot{x}}{\\partial \\dot{y}} \\\\ \n",
    "\\frac{\\partial \\ddot{x}}{\\partial x} & \n",
    "\\frac{\\partial \\ddot{x}}{\\partial \\dot{x}}& \n",
    "\\frac{\\partial \\ddot{x}}{\\partial y}& \n",
    "\\frac{\\partial \\dot{x}}{\\partial \\dot{y}}\\\\\n",
    "\\frac{\\partial \\dot{y}}{\\partial x} & \n",
    "\\frac{\\partial \\dot{y}}{\\partial \\dot{x}} & \n",
    "\\frac{\\partial \\dot{y}}{\\partial y} & \n",
    "\\frac{\\partial \\dot{y}}{\\partial \\dot{y}} \\\\ \n",
    "\\frac{\\partial \\ddot{y}}{\\partial x} & \n",
    "\\frac{\\partial \\ddot{y}}{\\partial \\dot{x}}& \n",
    "\\frac{\\partial \\ddot{y}}{\\partial y}& \n",
    "\\frac{\\partial \\dot{y}}{\\partial \\dot{y}}\n",
    "\\end{bmatrix}\\normalsize\n",
    "\\begin{bmatrix}\\Delta x \\\\ \\Delta \\dot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix}$$\n",
    "\n",
    "The equations do not contain both an x and a y, so any partial derivative with both in it must be equal to zero. We also know that $\\large\\frac{\\partial \\dot{x}}{\\partial x}\\normalsize = 0$ and that  $\\large\\frac{\\partial \\dot{x}}{\\partial \\dot{x}}\\normalsize = 1$, so our matrix ends up being\n",
    "\n",
    "$$\\mathbf{F} = \\begin{bmatrix}0&1&0&0 \\\\\n",
    "\\frac{0.0034e^{-x/22000}\\dot{x}^2g}{44000\\beta}&0&0&0\n",
    "\\end{bmatrix}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{aligned}\\ddot{x} &= -\\frac{1}{2}C_d\\rho A \\dot{x}\\\\\n",
    "\\ddot{y} &= -\\frac{1}{2}C_d\\rho A \\dot{y}-g\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.abc import *\n",
    "from sympy import *\n",
    "\n",
    "init_printing(pretty_print=True, use_latex='mathjax')\n",
    "\n",
    "x1 = (0.0034*g*exp(-x/22000)*((x)**2))/(2*b) - g\n",
    "\n",
    "x2 = (a*g*exp(-x/c)*(Derivative(x)**2))/(2*b) - g\n",
    "\n",
    "#pprint(x1)\n",
    "#pprint(Derivative(x)*Derivative(x,n=2))\n",
    "#pprint(diff(x2, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** orphan text\n",
    "This approach has many issues. First, of course, is the fact that the linearization does not produce an exact answer. More importantly, we are not linearizing the actual path, but our filter's estimation of the path. We linearize the estimation because it is statistically likely to be correct; but of course it is not required to be. So if the filter's output is bad that will cause us to linearize an incorrect estimate, which will almost certainly lead to an even worse estimate. In these cases the filter will quickly diverge. This is where the 'black art' of Kalman filter comes in. We are trying to linearize an estimate, and there is no guarantee that the filter will be stable. A vast amount of the literature on Kalman filters is devoted to this problem. Another issue is that we need to linearize the system using analytic methods. It may be difficult or impossible to find an analytic solution to some problems. In other cases we may be able to find the linearization, but the computation is very expensive. **\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[1] http://sympy.org\n"
   ]
  }
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